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1.5 Nested Quantifiers

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277 views2 pages

1.5 Nested Quantifiers

Công ty Kiếm thẻ cào chúng tôi chia sẻ đến bạn cách kiếm thẻ cào trên điện thoại Android và IOS giúp bạn dễ dàng kiếm thêm thu nhập mỗi ngày mà không mất bất kỳ chi phí nào từ nhiều hình thức khác nhau: làm khảo sát online, đọc báo hay 24h kiếm tiền. Address: Công Ty Kiếm Thẻ Cào Điện Thoại - Xóm Trung Tiến Diễn Hoàng - Diễn Châu - Nghệ An. Phone: (+84) 37 854 4082 Contact Us: Website: https://kiemthecaofree.com/ Facebook: https://www.facebook.com/kiemthecaocom

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[INT 1358]: Discrete Mathematics I

1.5 Nested Quantifiers


Nested quantifiers are quantifiers that occur within the scope of other quantifiers.
Example: ∀x∃yP (x, y)

Quantifier order matters!


∀x∃yP (x, y) 6= ∃y∀xP (x, y)

1.5 pg. 65 # 9
Let L(x, y) be the statement “x loves y,” where the domain for both x and y consists of all people
in the world. Use quantifiers to express each of these statements.

a) Everybody loves Jerry.


∀xL(x, Jerry)

b) Everybody loves somebody.


∀x∃yL(x, y)

c) There is somebody whom everybody loves.


∃y∀xL(x, y)

d) Nobody loves everybody.


∀x∃y¬L(x, y) or ¬∃x∀yL(x, y)

i Everyone loves himself or herself


∀xL(x, x)

1.5 pg. 64 # 5
Let W (x, y) mean that student x has visited website y, where the domain for x consists of all
students in your school and the domain for y consists of all websites. Express each of these
statements by a simple English sentence.

d ∃y(W (Ashok Puri,y) ∧ W (Cindy Yoon, y))


There is a website that both Ashok and Cindy have visited.

e ∃y∀z(y 6= (David Belcher) ∧ (W (David Belcher, z) → W (y, z)))


There is a person besides David who has visited all the websites that David has visited.

f ∃x∃y∀z(((x 6= y) ∧ (W (x, z) ↔ W (y, z))))


There are two distinct people who have visited exactly the same sites.

1
[INT 1358]: Discrete Mathematics I

1.5 pg. 66 # 13
Let M (x, y) be “x has sent y an e-mail message” and T (x, y) be “x has telephoned y,” where the
domain consists for all students in your class. Use quantifiers to express each of these statements.

k There is a student in your class who has not received an e-mail message from anyone else in
the class and who has not been called by any other student in the class.
∃x∀y((x 6= y) → (¬M (y, x) ∧ ¬T (y, x)))

l Every student in the class has either received an e-mail message or received a telephone call
from another student in the class.
∀x∃y((x 6= y) ∧ (M (y, x) ∨ T (y, x)))

m There are at least two students in your class such that one student has sent the other e-mail
and the second student has telephoned the first student
∃x∃y((x 6= y) ∧ (M (x, y) ∧ T (y, x)))

1.5 pg. 67 # 33
Rewrite each of these statements so that negations appear only within predicates (that is, so that no
negation is outside a quantifier or an expression involving logical connectives).

a) ¬∀x∀yP (x, y)
¬∀x∀yP (x, y)
≡ ∃x¬∀yP (x, y) De Morgan’s laws for quantifiers
≡ ∃x∃y¬P (x, y) De Morgan’s laws for quantifiers

d ¬(∃x∃y¬P (x, y) ∧ ∀x∀y(Q(x, y))


¬(∃x∃y¬P (x, y) ∧ ∀x∀y(Q(x, y))
≡ (¬∃x∃y¬P (x, y)) ∨ (¬∀x∀yQ(x, y)) De Morgan’s Law
≡ (∀x¬∃y¬P (x, y)) ∨ (¬∀x∀yQ(x, y)) De Morgan’s laws for quantifiers
≡ (∀x∀y¬¬P (x, y)) ∨ (¬∀x∀yQ(x, y)) De Morgan’s laws for quantifiers
≡ (∀x∀yP (x, y)) ∨ (¬∀x∀yQ(x, y)) Double Negation
≡ (∀x∀yP (x, y)) ∨ (∃x¬∀yQ(x, y)) De Morgan’s laws for quantifiers
≡ (∀x∀yP (x, y)) ∨ (∃x∃y¬Q(x, y)) De Morgan’s laws for quantifiers

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